Completeness of S4 for the Lebesgue Measure Algebra

Abstract

We prove completeness of the propositional modal logic S4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, $\Box$ and $\Diamond$. Propositional modal formulae are assigned to Lebesgue-measurable subsets of the real interval [0, 1], modulo sets of measure zero. Equivalence classes of Lebesgue-measurable subsets form a measure algebra, $\mathcal M$, and we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in $\mathcal M$ with an open representative. We prove completeness of the modal logic S4 for the algebra $\mathcal M$. A corollary to the main result is that non-theorems of S4 can be falsified at each point in a subset of the real interval [0, 1] of measure arbitrarily close to 1. A second corollary is that Intuitionistic propositional logic (IPC) is complete for the frame of open elements in $\mathcal M$.